Vincent Torre

Differential techniques for optical flow

We show that optical flow, i.e., the apparent motion of the time-varying brightness over the image plane of an imaging device, can be estimated by means of simple differential techniques. Linear algebraic equations for the two components of optical flow at each image location are derived. The coefficients of these equations are combinations of spatial and temporal derivatives of the image brightness. The equations are suggested by an analogy with the theory of deformable bodies and are exactly true for particular classes of motion or elementary deformations. Locally, a generic optical flow can be approximated by using a constant term and a suitable combination of four elementary deformations of the time-varying image brightness, namely, a uniform expansion, a pure rotation, and two orthogonal components of shear. When two of the four equations that correspond to these deformations are satisfied, optical flow can more conveniently be computed by assuming that the spatial gradient of the image brightness is stationary. In this case, it is also possible to evaluate the difference between optical flow and motion field—that is, the two-dimensional vector field that is associated with the true displacement of points on the image plane. Experiments on sequences of real images are reported in which the obtained optical flows are used successfully for the estimate of three-dimensional motion parameters, the detection of flow discontinuities, and the segmentation of the image in different moving objects.

Mathematical properties of the two-dimensional motion field: from singular points to motion parameters

The motion field, that is, the two-dimensional vector field associated with the velocity of points on the image plane, can be seen as the flow vector of the solution to a planar system of differential equations. Therefore the theory of planar dynamical systems can be used to understand qualitative and quantitative properties of motion. In this paper it is shown that singular points of the motion field, which are the points where the field vanishes, and the time evolution of their local structure capture essential features of three-dimensional motion that make it possible to distinguish translation, rotation, and general motion and also make possible the computation of the relevant motion parameters. Singular points of the motion field are the perspective projection onto the image plane of the intersection between a curve called the characteristic curve, which depends on only motion parameters, and the surface of the moving object. In most cases, singular points of the motion field are left unchanged in location and spatial structure by small perturbations affecting the vector field. Therefore a description of motion based on singular points can be used even when the motion field of an image sequence has not been estimated with high accuracy.

Constraints for the computation of optical flow

A number of constraints are proposed for which both the components of optical flow can be obtained by local differential techniques and the aperture problem can usually be solved. The constraints are suggested by the observation that it is possible to describe spatial and temporal changes of the image brightness in terms of infinitesimal deformations. An arbitrary choice of two of the four equations which correspond to the elementary deformations of a 2-D pattern implies that the spatial gradient of the image brightness is stationary and leads to a linear system of equations for optical flow which seems best suited for numerical implementation on real data in the absence of a priori information. In that case, the error term between the computed optical flow and the motion field-that is, the 2-D vector field associated with the true displacement of points on the image plane-is derived and the conditions under which it can safely be neglected are discussed. Experiments on real images are reported which show that the obtained optical flows allow the estimate of 3-D motion parameters, the detection of discontinuities in the flow field, and the segmentation of the image in different moving objects.<<ETX>>

Shape Recognition and Retrieval Using String of Symbols

In this paper we present two algorithms for shape recognition. Both algorithms map the contour of the shape to be recognized into a string of symbols. The first algorithm is based on supervised learning using string kernels as often used for text categorization and classification. The second algorithm is very weakly supervised and is based on the procrustes analysis and on the edit distance used for computing the similarity between strings of symbols. The second algorithm correctly recognizes 98.29% of shapes from the MPEG-7 database, i.e. better than any previous algorithms. The second algorithm is able also to retrieve similar shapes from a database

Shape categorization using string kernels

In this paper, a novel algorithm for shape categorization is proposed. This method is based on the detection of perceptual landmarks, which are scale invariant. These landmarks and the parts between them are transformed into a symbolic representation. Shapes are mapped into symbol sequences and a database of shapes is mapped into a set of symbol sequences and therefore it is possible to use support vector machines for categorization. The method here proposed has been evaluated on silhouettes database and achieved the highest recognition result reported with a score of 97.85% for the MPEG-7 shape database.

3D visual information from vanishing points

It is shown that when a priori information on the mutual direction of straight lines in the 3D scene is available, vanishing points can be extracted and located reliably on the image plane. Then, by using simple properties of vanishing points, it is possible to perform efficiently several visual tasks, such as estimating the rotational component of motion between the viewer and the scene, and identifying and recovering the orientation and relative position in space of the viewed planar patches. Extensive experimentation on real images shows that vanishing points are usually identified and located correctly, even in cluttered images. It is concluded that the proposed approach can be very helpful for the development of effective vision systems.<<ETX>>

A vision system for recognizing objects in complex real images

A new system for object recognition in complex natural images is here proposed. The proposed system is based on two modules: image segmentation and region categorization. Original images g(x,y) are first regularized by using a self-adaptive implementation of the Mumford-Shah functional so that the two parameters α and γ controlling the smoothness and fidelity, automatically adapt to the local scale and contrast. From the regularized image u(x,y), a piece-wise constant image sN(x,y) representing a segmentation of the original image g(x,y) is obtained. The obtained segmentation is a collection of different regions or silhouettes which must be categorized. Categorization is based on the detection of perceptual landmarks, which are scale invariant. These landmarks and the parts between them are transformed into a symbolic representation. Shapes are mapped into symbol sequences and a database of shapes in mapped into a set of symbol sequences. Categorization is obtained by using support vector machines. The Kimia silhouettes database is used for training and complex natural images from Martin database and collection of images extracted from the web are used for testing the proposed system. The proposed system is able to recognize correctly birds, mammals and fish in several of these cluttered images.

Unsupervised clustering of shapes

A new method for unsupervised clustering of shapes is here proposed. This method is based on two steps: in the first step a preliminary clusterization is obtained by considering the distance among shapes after alignment with procrustes analysis [1],[2]. This step is based on the minimization of the functional θ(N cluster ) =αN cluster + (1/N cluster )dist(c i ) where N cluster is the total number of clusters, dist(c i ) is the intra-cluster variability and α is an appropriate constant. In the second step, the curvature of shapes belonging to clusters obtained in the first step is examined to i) identify possible outliers and to ii) introduce a further refinement of clusters. The proposed method was tested on the Kimia, Surrey and MPEG7 shape databases and was able to obtain correct clusters, corresponding to perceptually homogeneous object categories. The proposed method was able to distinguish shapes with subtle differences, such as birds with one or two feet and to distinguish among very similar animal species....